Decay rates for a class of diffusive-dominated interaction equations

Details Decay rates for a class of diffusive-dominated interaction equations Decay rates for a class of diffusive-dominated interaction equations

2011-06-29

arxiv.org/abs/1106.5880v3

Journal of Mathematical Analysis and Applications 389(1):541-557 (May 2012)

Mathematics

Analysis of PDEs

22 pages

Scientific paper

10.1016/j.jmaa.2011.12.006

We analyse qualitative properties of the solutions to a mean-field equation for particles interacting through a pairwise potential while diffusing by Brownian motion. Interaction and diffusion compete with each other depending on the character of the potential. We provide sufficient conditions on the relation between the interaction potential and the initial data for diffusion to be the dominant term. We give decay rates of Sobolev norms showing that asymptotically for large times the behavior is then given by the heat equation. Moreover, we show an optimal rate of convergence in the $L^1$-norm towards the fundamental solution of the heat equation.

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